Image processing researchers commonly assert that "median filtering is betterthan linear filtering for removing noise in the presence of edges." Using astraightforward large-$n$ decision-theory framework, this folk-theorem is seento be false in general. We show that median filtering and linear filtering havesimilar asymptotic worst-case mean-squared error (MSE) when the signal-to-noiseratio (SNR) is of order 1, which corresponds to the case of constant per-pixelnoise level in a digital signal. To see dramatic benefits of median smoothingin an asymptotic setting, the per-pixel noise level should tend to zero (i.e.,SNR should grow very large). We show that a two-stage median filtering usingtwo very different window widths can dramatically outperform traditional linearand median filtering in settings where the underlying object has edges. In thistwo-stage procedure, the first pass, at a fine scale, aims at increasing theSNR. The second pass, at a coarser scale, correctly exploits the nonlinearityof the median. Image processing methods based on nonlinear partial differentialequations (PDEs) are often said to improve on linear filtering in the presenceof edges. Such methods seem difficult to analyze rigorously in adecision-theoretic framework. A popular example is mean curvature motion (MCM),which is formally a kind of iterated median filtering. Our results on iteratedmedian filtering suggest that some PDE-based methods are candidates torigorously outperform linear filtering in an asymptotic framework.
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